scholarly journals Exact solution of the planar motion of three arbitrary point vortices

2015 ◽  
Vol 29 (35n36) ◽  
pp. 1530017
Author(s):  
Robert Conte ◽  
Laurent de Seze
Keyword(s):  
Exact Solution ◽  
Equilibrium Configuration ◽  
Relative Equilibrium ◽  
Arbitrary Point ◽  
Triple Collision ◽  
Stability Conditions ◽  
Adiabatic Invariants ◽  
The Third ◽  
Elastic Diffusion ◽  
Equilateral Triangles

We give an exact quantitative solution for the motion of three vortices of any strength, which Poincaré showed to be integrable. The absolute motion of one vortex is generally biperiodic: in uniformly rotating axes, the motion is periodic. There are two kinds of relative equilibrium configuration: two equilateral triangles and one or three colinear configurations, their stability conditions split the strengths space into three domains in which the sets of trajectories are topologically distinct. According to the values of the strengths and the initial positions, all the possible motions are classified. Two sets of strengths lead to generic motions other than biperiodic. First, when the angular momentum vanishes, besides the biperiodic regime there exists an expansion spiral motion and even a triple collision in a finite time, but the latter motion is nongeneric. Second, when two strengths are opposite, the system also exhibits the elastic diffusion of a vortex doublet by the third vortex. For given values of the invariants, the volume of the phase space of this Hamiltonian system is proportional to the period of the reduced motion, a well known result of the theory of adiabatic invariants. We then formally examine the behaviour of the quantities that Onsager defined only for a large number of interacting vortices.

scholarly journals Reconfiguration Analysis of a 3-DOF Parallel Mechanism

Robotics ◽  
2019 ◽  
Vol 8 (3) ◽  
pp. 66
Author(s):  
Maurizio Ruggiu ◽  
Xianwen Kong
Keyword(s):  
Parallel Mechanism ◽  
Degrees Of Freedom ◽  
Operation Mode ◽  
Parallel Mechanisms ◽  
Universal Joint ◽  
Kinematic Chains ◽  
Constraint Equations ◽  
The Third ◽  
Equilateral Triangles ◽  
Moving Platform

This paper deals with the reconfiguration analysis of a 3-DOF (degrees-of-freedom) parallel manipulator (PM) which belongs to the cylindrical parallel mechanisms family. The PM is composed of a base and a moving platform shaped as equilateral triangles connected by three serial kinematic chains (legs). Two legs are composed of two universal (U) joints connected by a prismatic (P) joint. The third leg is composed of a revolute (R) joint connected to the base, a prismatic joint and universal joint in sequence. A set of constraint equations of the 1-RPU−2-UPU PM is derived and solved in terms of the Euler parameter quaternion (a.k.a. Euler-Rodrigues quaternion) representing the orientation of the moving platform and of the Cartesian coordinates of the reference point on the moving platform. It is found that the PM may undergo either the 3-DOF PPR or the 3-DOF planar operation mode only when the base and the moving platform are identical. The transition configuration between the operation modes is also identified.

Elastic anomalies of Hg2X2 compounds

1992 ◽  
Vol 70 (9) ◽  
pp. 745-751
Author(s):  
K. S. Viswanathan ◽  
J. C. Jeeja Ramani
Keyword(s):  
Elastic Constants ◽  
Fourth Order ◽  
Order Parameter ◽  
Zero Point ◽  
Stability Conditions ◽  
Point Energy ◽  
The Third ◽  
Limit Formula ◽  
The Stability ◽  
Elastic Anomalies

The anomalies of the second-, third-, and fourth-order elastic constants are considered for the phase transition of Hg2X2 type of compounds. Expressions are obtained for the equilibrium values of the order parameters in the ferroelastic phase from the stability conditions. The fluctuation in the order parameter is evaluated from the Landau–Khalatnikov equation. An expression is derived for the shift in the zero-point energy in the low-temperature ferroelastic phase and the specific heat anomaly. It is shown that these are proportional to (T − T)2 and (T − Tc), respectively. All the anomalies of the second-order elastic (SOE) constants are obtained from a single general formula, and relations among them are established. The temperature variation of the SOE constants in the limit [Formula: see text] is discussed. Similarly, expressions are derived for the anomalies of the third- and fourth-order elastic constants. In the limit [Formula: see text] it is shown that these constants diverge as [Formula: see text] and [Formula: see text], respectively.

Environmental Influences on the Development of Traditional Conservation in the South Pacific Region

1985 ◽  
Vol 12 (3) ◽  
pp. 217-230 ◽  
Author(s):  
Margaret D. Chapman
Keyword(s):  
Third World ◽  
Relative Equilibrium ◽  
Environmental Influences ◽  
South Pacific ◽  
Resource Depletion ◽  
The South ◽  
Island Populations ◽  
The Third ◽  
Traditional Societies ◽  
The Third World

There is an urgent need for improved understanding of conservation attitudes in the Third World because of the increasing rate of resource depletion that is now occurring in the countries involved. Although conservation practices by traditional societies in the Third World have received much attention from research workers, the fact that some practices are intentional and others inadvertent has been largely ignored. However, it is the motivation for these intentional conservation measures and the environmental influences on the people who apply them, which is crucial to understanding variations in conservation behaviour among traditional societies.Traditional conservation in the South Pacific was based on a complex system of resource-use taboos which prevented overexploitation in the limited island environment. These taboos contributed to the achievement during pre- European times of what appears from historical accounts to have been a state of relative equilibrium between island populations and their resources.Predictability and extremeness are two environmental factors which are thought to affect the development of conservational behaviour. Both these factors were examined in the light of traditional conservation in the South Pacific. Droughts and hurricanes are the two main sources of environmental unpredictability in the South Pacific, although the islands vary considerably in the degree to which they are affected by them. It was concluded that a distinction between real and perceived environmental predictability was necessary before one could fully understand the influence of predictability upon the development of conservational behaviour in the South Pacific.

Two-dimensional problems of electrohydrostatic stability

1972 ◽  
Vol 272 (1224) ◽  
pp. 331-359 ◽  
Keyword(s):  
Final Section ◽  
Equilibrium Configuration ◽  
Global Analysis ◽  
Conducting Fluid ◽  
Geometrical Parameters ◽  
Approximate Theory ◽  
Cylindrical Conductor ◽  
The Third ◽  
General Terms ◽  
The Stability

The main intention of this paper is to study the breakdown of equilibrium of a conducting fluid surface such as a membrane when placed in the field of a charged cylindrical conductor. The work is presented in four sections. The first section outlines an approximate theory relevant when the gap between the cylinder and surface is small in comparison with the radius of the cylinder. In the second section, this theory is applied to a number of related problems notably the stability of conducting fluid filaments suspended on parallel wires. In the third section we introduce an extension of the asymptotic analysis of the preceding sections which removes the restriction of the small gap requirement and may be applied to problems of section two which possess symmetry about a centre plane. In the final section, we present a global analysis of the stability of a membrane in the field of a cylindrical conductor. Here the problem is studied in general terms without restrictions on geometrical parameters, and the resulting general equations determining the equilibrium configuration of the membrane are solved numerically.

scholarly journals The Third Boundary Value Problem for a Loaded Thermal Conductivity Equation with a Fractional Caputo Derivative

2020 ◽  
pp. 52-64
Author(s):  
M. Kh. Beshtokov ◽  
M. Z. Khudalov
Keyword(s):  
Exact Solution ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Boundary Value Problem ◽  
Caputo Derivative ◽  
Boundary Value ◽  
Conductivity Equation ◽  
Partial Differential ◽  
The Third ◽  
Fractional Caputo Derivative

Recently, to describe various mathematical models of physical processes, fractional differential calculus has been widely used. In this regard, much attention is paid to partial differential equations of fractional order, which are a generalization of partial differential equations of integer order. In this case, various settings are possible.Loaded differential equations in the literature are called equations containing values of a solution or its derivatives on manifolds of lower dimension than the dimension of the definitional domain of the desired function. Currently, numerical methods for solving loaded partial differential equations of integer and fractional (porous media) orders are widely used, since analytical solving methods for solving are impossible.In the paper, we study the initial-boundary value problem for the loaded differential heat equation with a fractional Caputo derivative and conditions of the third kind. To solve the problem on the assumption that there is an exact solution in the class of sufficiently smooth functions by the method of energy inequalities, a priori estimates are obtained both in the differential and difference interpretations. The obtained inequalities mean the uniqueness of the solution and the continuous dependence of the solution on the input data of the problem. Due to the linearity of the problem under consideration, these inequalities allow us to state the convergence of the approximate solution to the exact solution at a rate equal to the approximation order of the difference scheme. An algorithm for the numerical solution of the problem is constructed.

scholarly journals Static form-finding of normal and defective catenaries based on the analytical exact solution of the tensile Euler–Bernoulli beam

2018 ◽  
Vol 233 (7) ◽  
pp. 691-700 ◽  
Author(s):  
Farzad Vesali ◽  
Mohammad Ali Rezvani ◽  
Habibolah Molatefi ◽  
Markus Hecht
Keyword(s):  
Exact Solution ◽  
Equilibrium Configuration ◽  
Field Measurements ◽  
Steady State Solution ◽  
Contact Wire ◽  
Proposed Model ◽  
Before And After ◽  
Novel Method ◽  
Static Form ◽  
Euler Bernoulli Beam

The aim of this research is to propose and develop an analytical exact solution for finding the static equilibrium configuration of a catenary before and after incurring defects such as tension loss or a broken dropper. The procedure includes considering the steady-state solution of the dynamic motion equation of the contact wire and the messenger cable. The wire and the cable are considered as tensile Euler–Bernoulli beams. The stiffness matrix of the beam is configured and is used to calculate the dropper's dead load. Progressively, a novel method is proposed to find the equilibrium configuration of the same catenary after the defect. The results prove that the tension loss in the messenger cable is more precarious than the tension loss in the contact wire. The broken dropper causes a significant sag in the sub-span and increases the static forces of the adjacent droppers. A comparison with field measurements justifies the accuracy of the results of the proposed model.

The Design of Multidimensional Sound Interfaces

1995 ◽  
Author(s):  
Michael Cohen ◽  
Elizabeth M. Wenzel
Keyword(s):  
User Interface ◽  
Graphical User Interface ◽  
User Interfaces ◽  
Electronic Mail ◽  
Arbitrary Point ◽  
3D Audio ◽  
The Third ◽  
Technology Platforms ◽  
Early Computer ◽  
Cultural Isolation

Early computer terminals allowed only textual I/O. Because the user read and wrote vectors of character strings, this mode of I/O (character-based user interface, or “CUI”) could be thought of as one-dimensional, 1D. As terminal technology improved, users could manipulate graphical objects (via a graphical user interface, or “GUI”) in 2D. Although the I/O was no longer unidimensional, it was still limited to the planar dimensionality of a CRT or tablet. Now there exist 3D spatial pointers and 3D graphics devices; this latest phase of I/O devices (Blattner, 1992; Blattner and Dannenberg, 1992; Robinett, 1992) approaches the way that people deal with “the real world.” 3D audio (in which the sound has a spatial attribute, originating, virtually or actually, from an arbitrary point with respect to the listener) and more exotic spatial I/O modalities are under development. The evolution of I/O devices can be roughly grouped into generations that also correspond to the number of dimensions. Representative instances of each technology are shown in Table 8-1. This chapter focuses on the italicized entries in the third-generation aural sector. Audio alarms and signals of various types have been with us since long before there were computers, but even though music and visual arts are considered sibling muses, a disparity exists between the exploitation of sound and graphics in interfaces. (Most people think that it would be easier to be hearing- than sight-impaired, even though the incidence of disability-related cultural isolation is higher among the deaf than the blind.) For whatever reasons, the development of user interfaces has historically been focused more on visual modes than aural. This imbalance is especially striking in view of the increasing availability of sound in current technology platforms. Sound is frequently included and utilized to the limits of its availability or affordability in personal computers. However, computer-aided exploitation of audio bandwidth is only beginning to rival that of graphics. General sound capability is slowly being woven into the fabric of applications. Indeed, some of these programs are inherently dependent on sound—voicemail, or voice annotation to electronic mail, teleconferencing, audio archiving—while other applications use sound to complement their underlying functionality.

scholarly journals DGJ Method for Linear and Nonlinear Third order Fractional Differential Equation

2018 ◽  
Author(s):  
Mahmut Modanli
Keyword(s):  
Differential Equation ◽  
Exact Solution ◽  
Fractional Differential Equation ◽  
Initial Boundary ◽  
Third Order ◽  
Partial Differential ◽  
Approximation Solution ◽  
The Third ◽  
Numerical Result ◽  
Fractional Differential

DGJ (Daftardar-Gejii-Jafaris) method is used to obtain numerical solution of the third order fractional differential equation. Providing the DGJ method converges, the approximate solution is a good and effective numerical result which is close to the exact solution or the exact solution. For this,the examples of the explaning the method are presented. The proposed method is implemented for the approximation solution of the third order nonlinear fractional partial differential equations. The method was shown to be unsuitable and inconsistent for an example of a nonlinear fractional partial differential equation depend on initial-boundary value conditions. The fact that these numerical results are not consistent can be explained by the fact that the method is not convergent.

Adiabatic Invariants and the “Third” Integral

1966 ◽  
Vol 7 (5) ◽  
pp. 788-797 ◽  
Author(s):  
George Contopoulos
Keyword(s):  
Adiabatic Invariants ◽  
The Third

An exact solution for the calendering analysis of a third-order fluid

2016 ◽  
Vol 33 (2) ◽  
pp. 124-141 ◽  
Author(s):  
Muhammad Sajid ◽  
Nasir Ali ◽  
Muhammad Asif Javed
Keyword(s):  
Exact Solution ◽  
Flow Characteristics ◽  
Sheet Thickness ◽  
Power Input ◽  
Perturbation Solution ◽  
Total Power ◽  
Third Order ◽  
Operating Variables ◽  
The Third ◽  
Fluid Parameter

This paper presents the exact solution for calendering a third-order fluid under lubrication approximation. The solution obtained is valid for all values of the third-order fluid parameter. This exact solution is compared to the perturbation solution. The results show that the perturbation solution is valid for very small values of the third-order fluid parameter. Therefore, no significant deviation from the corresponding results of the Newtonian fluid is observed with the perturbation solution. The interesting quantities for mechanically designing a calendering system such as the force separating the two rolls and total power input into both rolls are calculated and shown graphically for large third-order fluid parameter. The material’s rheological features modify the pressure, flow characteristics, and all other operating variables significantly. In fact, the nip region pressure increases with increasing third-order fluid parameter. The exiting sheet thickness, power input, and roll separating force also increase with increasing third-order fluid parameter.

Sign in / Sign up

Export Citation Format

Share Document